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        <title>Convergence</title>

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                <h1 id="title" titleSize="">
                    Convergence
                </h1>
            
            <h1 id="motivation--definition">Motivation &amp; Definition</h1>
<p>The idea of <em>convergence</em> of elements $x_\alpha$ in $X$ is to find a single element $x\in X$ that is ‘close’ to as many elements $x_\alpha$ as possible. This heuristic can be formalized in many ways; here, we focus on interpreting ‘closeness’ in the <a href=https://zhaoshenzhai.github.io/mathwiki/topological_space.md class="internalLink references" title="Topological Space" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/topological_space.md&#34;, {&#34;Date&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Topological Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/topological_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Topological Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/topological_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/topological_space.md&#34;, {&#34;Date&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-14T14:45:50-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Topological Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/topological_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">topological</a> sense.</p>
<br>
<p>  Throughout, let $X$ be a topological space. For each $x\in X$, let $\mc{N}_x$ denote the set of all <em>not necessarily open</em> neighborhoods of $x$, i.e., sets $A\subseteq X$ containing $x$ such that $x\in U\subseteq A$ for some open set $U$. A <em>neighborhood base</em> of $x$ is a subset $\mc{B}_x\subseteq\mc{N}_x$ such that every $U\in\mc{N}_x$ contains some $B\in\mc{B}_x$.</p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b><p>A sequence $(x_n)_{n\in\N}$ in $X$ is said to <em>converge</em> to some $x\in X$ $-$ and write $x_n\to x$ $-$ if for every neighborhood $U\in\mc{N}_x$, there is some $N\in\N$ such that $x_n\in U$ for all $n\geq N$ <span style="color:gray">(that is, $x_n\in U$ for all eventually $n\in\N$)</span>.</p>
<br>
<p>  We say that $x$ is a <em>limit point</em> of $x_n$ if every $U\in\mc{N}_x$ contains infinitely-many points in $x_n$.</p>
</div>

<p>One might hope to probe topological properties of $X$ (like compactness, continuity, closure, etc) by analyzing the convergence of sequences in $X$, but this is not fruitful for general topological spaces; in some sense, this is because sequences are ‘countable objects’, while topological spaces in general require uncountably-many bits to specify.</p>
<br>
<p>  One can fix this problem by either restricting the class of topological spaces in question, say to <a href=https://zhaoshenzhai.github.io/mathwiki/first_countable_space.md class="internalLink references ghostLink" title="first" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/first_countable_space.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/first_countable_space.md&#34;, &#34;nopPage&#34;);">first</a>/<a href=https://zhaoshenzhai.github.io/mathwiki/second_countable_space.md class="internalLink references" title="Second Countable Space" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/second_countable_space.md&#34;, 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onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-08-28T21:46:15-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-08-28T21:46:15-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-08-28T21:46:15-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Second Countable Space&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/second_countable_space&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, 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a ‘countable encoding’ of the topology by way of <a href=https://zhaoshenzhai.github.io/mathwiki/bases_for_topologies.md class="internalLink references" title="Bases for Topologies" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/bases_for_topologies.md&#34;, {&#34;Date&#34;:&#34;2024-07-11T14:22:11-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-07-11T14:22:11-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-07-11T14:22:11-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Bases for 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<h1 id="spaces-probable-by-sequences">Spaces Probable by Sequences</h1>
<h2 id="metric-spaces">Metric Spaces</h2>
<h1 id="convergence-of-nets-and-hahahugoshortcode16s6hbhb">Convergence of Nets and <a href=https://zhaoshenzhai.github.io/mathwiki/filter.md class="internalLink references" title="Filter" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/filter.md&#34;, {&#34;Date&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Filter&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/filter&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Filter&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/filter&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/filter.md&#34;, {&#34;Date&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Filter&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/filter&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">Filters</a></h1>
<p>The two most common approaches towards such a generalization are nets and filters, which are basically equivalent $-$ in that one can be defined from the other $-$ but it is worthwhile to understand both. For both of these generalizations, the neighborhoods $\mc{N}_x$ of a point $x\in X$, and there bases, play a fundamental role.</p>
<h2 id="nets">Nets</h2>
<p>One approach to replace $\N$ in the definition of sequential-convergence to an arbitrary directed set, which we call a <em>net</em>.</p>
<br>
<p>  Fix a net $x_\blob:I\to X$ from a directed set $I$. For a subset $A\subseteq X$, we say that $x_\blob$ is <em>eventually in</em> (resp. <em>cofinally in</em>) $A$ if there is some $j\in I$ such that $x_i\in A$ for all $i\geq j$ (resp. for all $j\in I$, there is some $i\geq j$ with $x_i\in A$). We make the following</p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b><p>Let $I$ be a directed set. A net $x_\blob:I\to X$ is said to <em>converge</em> to some $x\in X$ $-$ and write $x_\blob\to x$ $-$ if $x_\blob$ is eventually in every $U\in\mc{N}_x$.</p>
<br>
<p>  We say that $x$ is a <em>limit point</em> of $x_\blob$ if $x_\blob$ is cofinally in every $U\in\mc{N}_x$.</p>
</div>

<p>If $\mc{N}_x$ admits a countable base $\mc{B}_x\subseteq\mc{N}_x$ $-$ which can be nested $-$ then $\mc{N}_x$ admits a cofinal subset $\mc{B}_x$ order-isomorphic to $\N$. With $I\coloneqq\mc{B}_x$, this notion generalizes that of sequences $x_\blob:\N\to X$.</p>
<br>
<p>  Despite this simple generalization, the topology of $X$ is completely determined in terms of the convergence of nets in $X$. This is outlined here<a href=https://zhaoshenzhai.github.io/mathwiki/topology_using_nets.md class="internalLink properties ghostLink dag" title="" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/topology_using_nets.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/topology_using_nets.md&#34;, &#34;nopPage&#34;);">$^\dagger$</a>.</p>
<h2 id="filters">Filters</h2>
<p>Since pairwise-intersections of open sets are open, neighborhood bases of $\mc{N}_x$ are just filter bases thereof. Thus, we can compare it to other filters.</p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b><p>A filter $\mc{F}$ in $X$ is said to <em>converge</em> to some $x\in X$ $-$ and write $\mc{F}\to x$ $-$ if $\mc{F}$ refines $\mc{N}_x$ <span style="color:gray">(that is, if every $U\in\mc{N}_x$ contains some $A\in\mc{F}$)</span>.</p>
<br>
<p>  We say that $x$ is a <em>limit point</em> of $\mc{F}$ if every $A\in\mc{F}$ meets every $U\in\mc{N}_x$.</p>
</div>

<p><strong>Remark.</strong> One can replace ‘filter’ by ‘prefilter’ by passing to its generated filter.</p>
<div class="space"></div>
<p>  To connect this definition back with sequences, we need to consider <em><a href=https://zhaoshenzhai.github.io/mathwiki/filter.md/#Push%20Forward class="internalLink references" title="Filter" mathLink="" secID="Push Forward" secDisplay="push-forward" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/filter.md/#Push Forward&#34;, {&#34;Date&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Filter&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/filter&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Filter&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/filter&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/filter.md/#Push Forward&#34;, {&#34;Date&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T11:27:17-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Filter&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/filter&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">push-forward</a></em> of filters. Then, equipping $\N$ with the Fréchet filter $\mc{F}_0$, we see that a sequence $x_\blob\to x$ in the sense of sequences iff $x_\blob(\mc{F}_0)\to x$ in the sense of filters.</p>


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                June 20, 2024 | Zhaoshen Zhai

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